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Multiple correlations of spectra for higher rank Anosov representations

Michael Chow, Hee Oh

TL;DR

The paper develops a higher-rank framework for counting and equidistributing joint spectral data of Anosov representations by introducing hypertubes, higher-dimensional truncations in the Cartan (or Jordan) projection space. Key components are the existence of a unique $(\boldsymbol{\varphi},\boldsymbol{r})$-critical vector, joint equidistribution results with holonomies, and precise asymptotics for counting in hypertubes, expressed through the growth indicator $\psi_{\boldsymbol{\rho}(\Sigma)}$ evaluated at $\mathsf v^{\star}$. The main outcome is a universal formula: for interior $\boldsymbol{r}$, the counts of $[\sigma]$ with $\boldsymbol{\varphi}(\lambda(\boldsymbol{\rho}(\sigma)))$ in a box and holonomies in a fixed set scale like $c\,e^{\delta T}/T^{(d+1)/2}$ for Jordan data and $c'\,c\,e^{\delta T}/T^{(d-1)/2}$ for Cartan data, with $\delta=\psi_{\boldsymbol{\rho}(\Sigma)}(\mathsf v^{\star})$, and an analogous upper bound. These results generalize earlier rank-one correlations to higher rank and unify counting and equidistribution via hypertubes, with implications for convex projective and Hitchin-type structures under deformation.

Abstract

We describe multiple correlations of Jordan and Cartan spectra for any finite number of Anosov representations of a finitely generated group. This extends our previous work on correlations of length and displacement spectra for rank one convex cocompact representations. Examples include correlations of the Hilbert length spectra for convex projective structures on a closed surface as well as correlations of eigenvalue gaps and singular value gaps for Hitchin representations. We relate the correlation problem to the counting problem for Jordan and Cartan projections of an Anosov subgroup with respect to a family of carefully chosen truncated {\it hypertubes}, rather than in tubes as in our previous work. Hypertubes go to infinity in a linear subspace of directions, while tubes go to infinity in a single direction and this feature presents a novel difficulty in this higher rank correlation problem.

Multiple correlations of spectra for higher rank Anosov representations

TL;DR

The paper develops a higher-rank framework for counting and equidistributing joint spectral data of Anosov representations by introducing hypertubes, higher-dimensional truncations in the Cartan (or Jordan) projection space. Key components are the existence of a unique -critical vector, joint equidistribution results with holonomies, and precise asymptotics for counting in hypertubes, expressed through the growth indicator evaluated at . The main outcome is a universal formula: for interior , the counts of with in a box and holonomies in a fixed set scale like for Jordan data and for Cartan data, with , and an analogous upper bound. These results generalize earlier rank-one correlations to higher rank and unify counting and equidistribution via hypertubes, with implications for convex projective and Hitchin-type structures under deformation.

Abstract

We describe multiple correlations of Jordan and Cartan spectra for any finite number of Anosov representations of a finitely generated group. This extends our previous work on correlations of length and displacement spectra for rank one convex cocompact representations. Examples include correlations of the Hilbert length spectra for convex projective structures on a closed surface as well as correlations of eigenvalue gaps and singular value gaps for Hitchin representations. We relate the correlation problem to the counting problem for Jordan and Cartan projections of an Anosov subgroup with respect to a family of carefully chosen truncated {\it hypertubes}, rather than in tubes as in our previous work. Hypertubes go to infinity in a linear subspace of directions, while tubes go to infinity in a single direction and this feature presents a novel difficulty in this higher rank correlation problem.
Paper Structure (6 sections, 22 theorems, 154 equations, 2 figures)

This paper contains 6 sections, 22 theorems, 154 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on discrete subgroups of $G$
  3. Critical vectors for proper linear maps
  4. Hypertubes and multiple correlations
  5. Equidistribution with respect to hypertubes
  6. Multiple correlations for Anosov representations

Key Result

Theorem 1.1

Let $d\ge 2$ and $\boldsymbol{\rho} = (\rho_1,\dots,\rho_d):\Gamma\to \operatorname{PGL}_3(\mathbb R)$ be an independent $d$-tuple of Zariski dense strongly convex cocompact projective representations. Then for any interior vector $\boldsymbol{r} =(r_1,\dots, r_d) \in \mathcal{L}_{\boldsymbol{\rho}} for some $c>0$. Moreover, we have the upper bound: which is strict when all $\delta_{\rho_i}r_i$ a

Figures (2)

  • Figure 1: Tube vs. hypertube.
  • Figure 2: A hypertube of $\mathcal{L}$.

Theorems & Definitions (43)

  • Theorem 1.1: Multiple correlations of Hilbert length spectra
  • Remark 1.2
  • Theorem 1.3: Multiple correlations of eigenvalue and singular value gaps
  • Remark 1.4
  • Theorem 1.5: Multiple correlations of Jordan and Cartan spectra
  • Remark 1.6
  • Theorem 2.1: KMO21
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • ...and 33 more